To begin with, we start from the ML estimation of the uncoded FTN system.
Let us consider that the error sequence \(\mathbf {e}\)  has a Hamming error weight \(w\) .
Therefore, it is obvious that there are in total \(\mathcal {A}=\binom{N}{w}2^w\)  combinations for \(\mathbf {e}\) .
For the convenience of notation, we use \(\mathbf {e}^{(w,j)} \in \mathcal {S}^{(N \times 1)}\)  to denote a specific error sequence \(\mathbf {e}\)  with a Hamming error weight \(w\) , where \(j \in \lbrace 1,...,\mathcal {A} \rbrace \)  indicates the \(j\) -th instance of all possible combinations of \(\mathbf {e}\) .
Then, based on Lemma 1, the BER \(P_b\)  can be upper-bounded by applying the union bound [1]}, such as:

Pb < w = 1N jA w2wNQ(d2(e(w,j))EbN0 )  ,

where

Q()=12 ( - 2/2)d ,

is the well-known Q-function [1]}.
